Let $H$ be a subspace of $V$.
For $c\in V,$ define $E(c) = \{c + h\,|\,h\in H\}$
Let $Q = \{ E(v)\,|\,v\in V\}$.
Define addition in $Q$ by: for $v, w\in V$, $E(v) \bigoplus E(w) = E(v+w)$
Define scalar multiplication in $Q$ by: for $v\in V$ and $\propto \in \mathbb{R}, \propto E(v) = E(\propto v)$.
Show that $Q$ together with this addition and scalar multiplication is a vector space.
My Attempt:
To check that $Q$ is a vector space, one must check each of the 10 axioms of a vector space to see if they hold.
$A_1:$ Let $a, b\in Q$. Then \begin{align*} a + b = E(a) + E(b) \in Q \end{align*} Therefore $Q$ is closed under addition ($A_1$ holds).
I haven't completed the whole solution yet, but you could imagine how long it would take. Is there a shorter and better way of proving?